The Cone of Hilbert Functions in the non-standard graded Case

TL;DR

This paper characterizes the cone of Hilbert functions for artinian graded modules over a polynomial ring with a non-standard grading, extending classical results to more general grading schemes.

Contribution

It provides a description of the cone of Hilbert functions in the non-standard graded case for modules over a polynomial ring in two variables.

Findings

Explicit description of the cone of Hilbert functions for non-standard grading

Extension of classical graded Hilbert function results to non-standard cases

Applicable to modules generated in degree 0 over R = k[x,y] with deg(y) = n

Abstract

We describe the cone of Hilbert functions of artinian graded modules finitely generated in degree 0 over the polynomial ring R = k[x, y] with the non-standard grading deg(x) = 1 and deg(y) = n, where n is any natural number.